The invention relates generally to improving the measurement accuracy of a scanning force microscope (SFM) and more particularly to determining the resonant frequencies in a cantilever such as is used in an SFM.
Scanning force microscopes are used in a broad range of fields. They can provide surface information at a very high resolution. A subset of SFMs utilize a flexible cantilever attached to a probe. FIG. 1 (Prior Art) shows one such SFM. The SFM includes a cantilever 100 and a tip 102, which are manufactured as a single piece of, typically, SiN3 or C. The cantilever 100 measures about 100 xcexcm in length, and the tip, made as regular tetrahedron, averages 10 xcexcm in height. A transducer scans the tip 102 across the surface 104 under study and, meanwhile, the tip 102 interacts with the surface 104 via a variety of microscopic forces.
Tip 102 reflects a laser beam 106 to a group of photodiodes 108. The deflection or movement of the tip 102 is measured by detecting movement of the reflected laser beam 106.
The function of the cantilever 100 is to support the tip 102. The tip 102 thus plays the main role of the microscope. The cantilever 100, in turn, purports an ancillary purpose. However, any reconstruction algorithm must incorporate it as a pivotal element. A reconstruction algorithm converts experimental data into force information. More specifically, the sample rests on a vertically movable support. As the support moves, so do the tip-sample separation, and their interaction force. The objective of a xe2x80x9cspectroscopicxe2x80x9d SFM consists of retrieving the corresponding force-separation (that is force vs. tip-sample-separation) curve. On the other hand, experiments measure kinematic data. By means of the photodiode system and simple geometrical optics, the height of the tip 102 as a function of time, z(t), is recorded through photodiodes 108. Thus, any reconstruction algorithm reduces the kinematics into the force-separation curve. These two pieces of information are linked through the dynamics of the cantilever-tip system under the influence of the sought force.
An earlier dynamics model that is still used extensively in analyzing data from an SFM is shown in FIG. 2. This model This model assumes that the elastic properties of the cantilever may be lumped into an effective spring constant, k, and the cantilever-tip inertia is considered through an effective mass, m. By this model spring 200 connects with the tip 202 that in turn interacts with a surface. As the sample-holder platform 204 moves upward at constant speed v, the tip-sample separation changes, and so does the tip-sample interaction. Within this framework and, by using Newton""s second law, the interaction is F(t)=md2z(t)/dt2+kz(t), where t represents the time elapsed since the platform started moving and, z(t), the tip""s height is measured with the optical setup shown in FIG. 1.
While this model provides relatively good results, the accuracy of an SFM is limited by the underlying model. Improvements to this model are taught in U.S. Pat. No. 6,145,374 and in U.S. patent application Ser. No. 09/545,570, both to Zypman et al., which are incorporated herein by reference.
The interactions of a cantilever such as those used in a SFM are mathematically modeled as system having multiple resonant frequencies. Due to the small size of a cantilever actually used in an SFM, verification of any mathematical model are difficult, at best. To overcome that difficulty, a macroscopic cantilever is disclsoed along with effective methods of determing resonant frequencies of the cantilever system. According to one preferred aspect of the invention, the cantilever is on the order of at least one centimeter. Although the system is developed for determining the accuracy of mathematical models used in SFMs, preferred embodiments of the invetion may also be used to characterize the physical properties of materials such as Young Modulus. The methods may be used with britle materials such as ceramics or glass. In addition, non-linear elastic properties can be studied. This is of particular relevance in predicting the behavior of materials when performing as parts of mobile machinery.
According to one aspect of the invention, the accuracy of an SFM is improved based upon multiple resonance frequencies of a cantilever system. The cantilever is composed of a material having a known Young""s modulus, E, and the the cross sectional area, A, length, L, and geometric moment of inertia, I, of the cantilever are determined. The vibrational modes of the cantilever system are calculated based upon the following equations:                               1          +                      cos            ⁢                          xe2x80x83                        ⁢                                          ξ                n                            ·              cosh                        ⁢                          xe2x80x83                        ⁢                          ξ              n                                                            sin            ⁢                          xe2x80x83                        ⁢                                          ξ                n                            ·              cosh                        ⁢                          xe2x80x83                        ⁢                          ξ              n                                -                      cos            ⁢                          xe2x80x83                        ⁢                                          ξ                n                            ·              sinh                        ⁢                          xe2x80x83                        ⁢                          ξ              n                                          ⁢              ξ        n        3              =    β              n      n        =                            (                                    A                              1                2                                      ⁢            V                    )                /                  (                      2            ⁢                          pL              2                                )                    ⁢              x        n        2            
where:
xcex2=GL3/EI; and 
G=slope of a forcexe2x88x92distance curve 
The SFM is electronically calibrated based upon these vibrational modes. According to a further aspect of the invention, the cantilever is excited by a piezoelectric crystal positioned near its base. Another piezoelectric crystal detects the vibrations at the free end of the cantilever. The excitation and detected signals are compared to determine the resonance frequencies of the system.
According to another aspect of the invention, the resonant frequencies of a cantilever system are determined. The cantilever has a length of at least one centimeter. An excitation is applied near the base of the cantilever and the displacement is measured through a transducer near the free end of the cantilever. The excitation signal is plotted against the detected signal. When a resonance frequency is generated, the plot will display a Lissajous figure.
According to a further aspect of the invention, the cantilever has a known spring constant, k. The slope of a force separation curve is then determined based upon the following equation:             1      3        ⁢                  1        +                  cos          ⁢                      xe2x80x83                    ⁢                      ξ            ·            cosh                    ⁢                      xe2x80x83                    ⁢          ξ                                      sin          ⁢                      xe2x80x83                    ⁢                      ξ            ·            cosh                    ⁢                      xe2x80x83                    ⁢          ξ                -                  cos          ⁢                      xe2x80x83                    ⁢                      ξ            ·            sinh                    ⁢                      xe2x80x83                    ⁢          ξ                      ⁢          ξ      3        =      k    κ  
According to another aspect of the invention, a measurement device is configured to detect the resonant frequencies of a cantilever. The measurement device includes a base, a cantilever, a pair of transducers and a display. The cantilever is attached to the base on one end and free, at the other end. The cantilever is at least on centimeter long. One of the transducers is positioned at the base and the other at the free end of the cantilever. A signal generator drives the transducer at the base at a single frequency that sweeps through a range. That excitation signal and the detected signal from the transducer at the free end are plotted on the display. According to a further aspect of the invention, the excitation is plotted against the detected signal so that resonant frequencies of the cantilever system generate a Lissajous figure. The resonat frequency is determined by a frequency counter connected with the signal generator.